U.S. patent number 10,968,827 [Application Number 15/278,477] was granted by the patent office on 2021-04-06 for anti-icing apparatus for a nose cone of a gas turbine engine.
This patent grant is currently assigned to PRATT & WHITNEY CANADA CORP.. The grantee listed for this patent is Pratt & Whitney Canada Corp.. Invention is credited to Daniel Alecu, Ivan Sidorovich Paradiso.
United States Patent |
10,968,827 |
Alecu , et al. |
April 6, 2021 |
Anti-icing apparatus for a nose cone of a gas turbine engine
Abstract
A fan nose cone is disclosed for impeding icing and recovering
momentum in a gas turbine engine. The fan nose cone comprises: an
axially symmetric shell having a convex external surface and an
internal surface, the shell having an opening in a forward end of
the shell for communication with a source of heated pressurized
air; and an axially symmetric deflector disposed forward of the
opening, the deflector being configured to direct heated
pressurized air exiting from the opening radially outwardly to flow
in a downstream direction over the convex external surface of the
shell during operation. The shell of the fan nose cone may have a
rearward circumferential vent in communication with the source of
heated pressurized air for directing heated pressurized from the
vent in a radially outward and downstream direction toward the fan
blade platforms.
Inventors: |
Alecu; Daniel (Brampton,
CA), Sidorovich Paradiso; Ivan (Toronto,
CA) |
Applicant: |
Name |
City |
State |
Country |
Type |
Pratt & Whitney Canada Corp. |
Longueuil |
N/A |
CA |
|
|
Assignee: |
PRATT & WHITNEY CANADA
CORP. (Longueuil, CA)
|
Family
ID: |
1000005468935 |
Appl.
No.: |
15/278,477 |
Filed: |
September 28, 2016 |
Prior Publication Data
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|
|
|
Document
Identifier |
Publication Date |
|
US 20180087456 A1 |
Mar 29, 2018 |
|
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
F02C
7/047 (20130101); F02C 7/00 (20130101); F01D
9/02 (20130101); F04D 19/002 (20130101); F05D
2220/32 (20130101) |
Current International
Class: |
F02C
7/047 (20060101); F01D 9/02 (20060101); F02C
7/00 (20060101); F04D 19/00 (20060101) |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
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622778 |
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May 1949 |
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GB |
|
634267 |
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Mar 1950 |
|
GB |
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1210202 |
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Oct 1970 |
|
GB |
|
Other References
Norton Rose Fulbright Canada LLP, Response to Office Action dated
Jan. 30, 2019 re: application No. 2,978,155. cited by applicant
.
Canadian Intellectual Property Office, Office Action dated Jul. 30,
2018 re: patent application No. 2,978,155. cited by
applicant.
|
Primary Examiner: Bomberg; Kenneth
Assistant Examiner: Getachew; Julian B
Attorney, Agent or Firm: Norton Rose Fulbright Canada
LLP
Claims
We claim:
1. A fan nose cone for a gas turbine engine having an axis of
rotation and a forward end relative to a primary airflow path
through the engine, the fan nose cone comprising: an axially
symmetric shell having a convex external surface and an internal
surface, the shell having an opening in a forward end of the shell,
the opening adapted to be in communication with a source of heated
pressurized air when the nose cone is installed on the engine; and
an axially symmetric deflector disposed forward of the opening in
the shell, the deflector having a rearward surface disposed forward
of and cooperating with the convex external surface of the shell to
define an annular air flow channel therebetween for directing
heated pressurized air exiting from the opening, the rearward
surface configured to radially outwardly direct said heated
pressurized air to flow in a downstream direction over the convex
external surface of the shell, the deflector having a central
protrusion that is rearwardly convex and that merges radially
outwardly with a concave rearward surface of revolution that
terminates at a circumferential edge of the deflector.
2. The fan nose cone according to claim 1 wherein the opening
comprises a single central opening.
3. The fan nose cone according to claim 1 wherein the deflector has
a convex forward surface of revolution that extends to the
circumferential edge.
4. The fan nose cone according to claim 1 wherein the shell has a
rearward circumferential vent in communication with the source of
heated pressurized air for directing heated pressurized air in a
radially outward and downstream direction.
5. The fan nose cone according to claim 4 wherein the vent is
disposed upstream of a plurality of fan blade platforms.
6. The fan nose cone according to claim 1 wherein the deflector is
configured to direct the heated pressurized air at least partially
rearward relative to the shell.
7. The fan nose cone according to claim 1, wherein the central
protrusion of the deflector extends into the opening of the
shell.
8. The fan nose cone according to claim 1, wherein a rearward
extremity of the central protrusion is rounded.
9. The fan nose cone according to claim 1, wherein a portion of the
convex external surface of the shell extending axially from within
the annular air flow channel to downstream of the annular air flow
channel has a constant radius of curvature.
10. The fan nose cone according to claim 1, wherein: the central
protrusion of the deflector extends into the opening of the shell;
a rearward extremity of the central protrusion is rounded; and a
portion of the convex external surface of the shell extending
axially from within the annular air flow channel to downstream of
the annular air flow channel has a constant radius of
curvature.
11. A gas turbine engine having a fan mounted on a shaft for
rotation about a fan axis, the fan comprising a fan hub supporting
a plurality of fan blades, and a fan nose cone comprising: an
axially symmetric shell having a convex external surface and an
internal surface, the shell having an opening in a forward end of
the shell, the opening adapted to be in communication with a source
of heated pressurized air in the engine; and an axially symmetric
deflector disposed forward of the opening in the shell, the
deflector having a rearward surface disposed forward of and
cooperating with the convex external surface of the shell to define
an annular air flow channel therebetween for directing heated
pressurized air exiting from the opening, the rearward surface
configured to radially outwardly direct said heated pressurized air
to flow in a downstream direction over the convex external surface
of the shell, the deflector having a central protrusion that is
rearwardly convex and that merges radially outwardly with a concave
rearward surface of revolution that terminates at a circumferential
edge of the deflector.
12. The gas turbine engine according to claim 11 wherein the
opening comprises a single central opening.
13. The gas turbine engine according to claim 11 wherein the
deflector has a convex forward surface of revolution that extends
to the circumferential edge.
14. The gas turbine engine according to claim 11 wherein the shell
has a rearward circumferential vent in communication with the
source of heated pressurized air for directing heated pressurized
air in a radially outward and downstream direction.
15. The gas turbine engine according to claim 14 wherein the vent
is disposed upstream of a plurality of fan blade platforms.
16. The gas turbine engine according to claim 11 wherein the
deflector is configured to direct the heated pressurized air at
least partially rearward relative to the shell.
17. The gas turbine engine according to claim 11, wherein the fan
nose cone is mounted for rotation with the shaft.
18. A method of impeding icing on a fan nose cone of a gas turbine
engine where the fan nose cone comprises an axially symmetric shell
having a convex external surface and an internal surface, the shell
having an opening in a forward end of the shell in communication
with a source of heated pressurized air in the engine, and an
axially symmetric deflector disposed forward of the opening, the
deflector having a rearward surface disposed forward of and
cooperating with the convex external surface of the shell to define
an annular air flow channel therebetween, the deflector having a
central protrusion that is rearwardly convex and that merges
radially outwardly with a concave rearward surface of revolution
that terminates at a circumferential edge of the deflector, the
method comprising: receiving heated pressurized air inside the fan
nose cone and allowing the heated pressurized air to exit via the
opening in the shell; and using the central protrusion and the
concave rearward surface of revolution to direct the heated
pressurized air exiting via the opening radially outwardly to flow
in a downstream direction over the convex external surface of the
shell.
19. The method according to claim 18, wherein the shell has a
rearward circumferential vent in communication with the source of
heated pressurized air, the method comprising: directing heated
pressurized air from the vent in a radially outward and downstream
direction.
20. The method according to claim 19 wherein the vent is disposed
upstream of a plurality of fan blades.
Description
TECHNICAL FIELD
The disclosure relates generally to aircraft engines and, more
particularly, to anti-icing of a fan nose cone.
BACKGROUND OF THE ART
An ice build-up on the outer surface of a fan nose cone of a gas
turbine engine can occur when an air flow containing moisture or
precipitation encounters the fan nose cone under appropriate
conditions. For example, ice can form when the air pressure,
humidity, air flow temperature and temperature of the fan nose cone
are within a specific range. Ice can accumulate in layers on the
fan nose cone and then can be dislodged by air flow and motion.
Hard ice particles flowing into the engine can cause foreign object
impact damage to blades and ducts.
Some prior art systems to prevent ice build-up include the use of
heated liquids and gases passing through channels in the fan nose
cone. Examples are shown in U.S. Pat. No. 8,015,789 to Brand et al
and in U.S. Pat. No. 8,210,825 to Jensen et al.
SUMMARY
In one aspect, the disclosure describes a fan nose cone for a gas
turbine engine having an axis of rotation and a forward end
relative to a primary airflow path through the engine. The fan nose
cone comprises:
an axially symmetric shell having a convex external surface and an
internal surface, the shell having an opening in a forward end of
the shell, the opening adapted to be in communication with a source
of heated pressurized air when the nose cone is installed on the
engine; and
an axially symmetric deflector disposed forward of the opening in
the shell, the deflector having a rearward surface disposed forward
of and cooperating with the convex external surface of the shell to
define an annular air flow channel therebetween for directing
heated pressurized air exiting from the opening, the rearward
surface configured to radially outwardly direct said heated
pressurized air to flow in a downstream direction over the convex
external surface of the shell.
The opening may comprise a single central opening.
The opening may comprise a plurality of apertures symmetrically
disposed about a central axis of the fan nose cone.
The deflector may have a central rearward convex protrusion that
merges radially outwardly with a concave rearward surface of
revolution that terminates at a circumferential edge of the
deflector.
The deflector may have a convex forward surface of revolution
merging at the circumferential edge.
The shell may have a rearward circumferential vent in communication
with the source of heated pressurized air for directing heated
pressurized air in a radially outward and downstream direction.
The vent may be disposed upstream of a plurality of fan blade
platforms.
The deflector may be configured to direct the heated pressurized
air at least partially rearward relative to the shell.
Embodiments may include combinations of the above features.
In another aspect, the disclosure describes a gas turbine engine
having a fan mounted on a shaft for rotation about a fan axis. The
fan comprises a fan hub supporting a plurality of fan blades, and a
fan nose cone comprising:
an axially symmetric shell having a convex external surface and an
internal surface, the shell having an opening in a forward end of
the shell, the opening adapted to be in communication with a source
of heated pressurized air in the engine; and
an axially symmetric deflector disposed forward of the opening in
the shell, the deflector having a rearward surface disposed forward
of and cooperating with the convex external surface of the shell to
define an annular air flow channel therebetween for directing
heated pressurized air exiting from the opening, the rearward
surface configured to radially outwardly direct said heated
pressurized air to flow in a downstream direction over the convex
external surface of the shell.
The opening may comprise a single central opening.
The opening may comprise a plurality of apertures symmetrically
disposed about the fan axis.
The deflector may have a central rearward convex protrusion that
merges radially outwardly with a concave rearward surface of
revolution that terminates at a circumferential edge.
The deflector may have a convex forward surface of revolution
merging at the circumferential edge.
The shell may have a rearward circumferential vent in communication
with the source of heated pressurized air for directing heated
pressurized air in a radially outward and downstream direction.
The vent may be disposed upstream of a plurality of fan blade
platforms.
The deflector may be configured to direct the heated pressurized
air at least partially rearward relative to the shell.
Embodiments may include combinations of the above features.
In a further aspect, the disclosure describes a method of impeding
icing on a fan nose cone of a gas turbine engine where the fan nose
cone comprises an axially symmetric shell having a convex external
surface and an internal surface, the shell having an opening in a
forward end of the shell in communication with a source of heated
pressurized air in the engine, and an axially symmetric deflector
disposed forward of the opening, the deflector having a rearward
surface disposed forward of and cooperating with the convex
external surface of the shell to define an annular air flow channel
therebetween, the method comprising:
receiving heated pressurized air inside the fan nose cone and
allowing the heated pressurized air to exit via the opening in the
shell; and
directing the heated pressurized air exiting via the opening
radially outwardly to flow in a downstream direction over the
convex external surface of the shell.
The shell may have a rearward circumferential vent in communication
with the source of heated pressurized air, the method directing
heated pressurized air from the vent in a radially outward and
downstream direction.
The vent may be disposed upstream of the plurality of fan
blades.
Embodiments may include combinations of the above features.
Further details of these and other aspects of the subject matter of
this application will be apparent from the detailed description
included below and the drawings.
DESCRIPTION OF THE DRAWINGS
Reference is now made to the accompanying drawings, in which:
FIG. 1 is an axial cross-section view through an exemplary
turbo-fan gas turbine engine having a fan with a fan nose cone with
air flow from left to right as drawn.
FIG. 2 is a front isometric view of the fan of FIG. 1 with multiple
blades with blade platforms forming a downstream air flow path and
a fan nose cone in accordance with the embodiment described
herein.
FIG. 3 is a detail front-left view of the front tip of the fan nose
cone showing a deflector for directing heated pressurized air from
a central opening to form a heated pressurized air curtain over the
convex external surface of the shell flowing in a radially outward
and downstream direction.
FIG. 4 is an axial sectional view through the front tip of the fan
nose cone of FIG. 3.
FIG. 5 is a detail front-left view of a circumferential vent
located at the downstream edge of the fan nose cone in
communication with the source of heated pressurized air for
directing a heated pressurized air curtain from the vent in a
radially outward and downstream direction toward the fan blade
platforms.
FIG. 6 is an axial sectional view through the downstream edge of
the fan nose cone of FIG. 5.
Further details of the invention and its advantages will be
apparent from the detailed description included below.
DETAILED DESCRIPTION
FIG. 1 illustrates a gas turbine engine 10 of a type preferably
provided for use in subsonic flight, generally comprising in serial
flow communication a fan 2 through which ambient air is propelled,
a multistage compressor 4 for pressurizing the air, a combustor 6
in which the compressed air is mixed with fuel and ignited for
generating an annular stream of hot combustion gases, and a turbine
section 11 for extracting energy from the combustion gases. Air
received into a forward end of engine 10 relative to a primary
airflow path passes axially over the fan nose cone 12 and the fan
blades 1 of the fan 2. The air received is then split into an outer
annular flow through the bypass duct 3 and an inner/core flow
through compressor 4.
Engine 10 may be of the type suitable for aircraft applications. It
will be understood that aspects of the disclosure may be equally
applicable to any type of engine with a fan nose cone and a source
of relatively hot air (e.g., from compressor 4). Engine 10 may
comprise apparatus 5 useful in providing anti-icing to fan nose
cone 12.
FIGS. 2-4 illustrate apparatus 5 for providing anti-icing to fan
nose cone 12. In some embodiments, apparatus 5 may comprise
deflector 22 configured to direct heated pressurized air exiting
from the opening 21 radially outwardly to flow as an air curtain
over the convex external surface of the shell in a downstream
direction during operation. Deflector 22 may also be configured to
direct the heated air partially rearwardly (i.e., toward the
downstream direction) during operation. In some embodiments,
deflector 22 may be in the form of a plug disposed upstream of and
partially occluding the opening 21. Pressurized heated air for
anti-icing purposes may originate from compressor 4 for example or
any other suitable location within engine 10. In some embodiments,
such pressurized heated air may be routed from compressor 4 to the
interior of fan nose cone 12 via one or more bearing housing
chambers to provide cooling to such bearings before getting to fan
nose cone 12. In some embodiments, such pressurized heated air may
be routed from compressor 4 to the interior of fan nose cone 12 via
a hollow center of low pressure shaft 7 to which fan 2 may be
drivingly coupled.
The flow of air exiting (i.e., being discharged from) opening 21
and being directed by deflector 22 may form a heated air curtain or
boundary layer flow that attaches to the exterior surface of fan
nose cone 12. For example, in some embodiments, the exterior convex
curvature of fan nose cone 12 and shape of deflector 22 facing
opening 21, through which heated air is ejected, may combine to
produce a Coanda effect or laminar air flow that maintains contact
with the convex exterior surface of fan nose cone 12 to improve the
anti-icing effectiveness provided by the heated air flow. Such
Coanda effect may be a tendency of a jet of fluid emerging from an
orifice to follow an adjacent curved surface and may entrain fluid
from the surroundings so that a region of lower pressure may
develop.
In some embodiments, fan nose cone 12 may comprise a substantially
axially symmetric hollow shell having an external surface exposed
to a free stream of air entering engine 10. In some embodiments, at
least part of the external surface of fan nose cone 12 may be
convex. The interior (i.e., internal surface) of fan nose cone 12
and hence opening 21 may be communication with the source of heated
pressurized air in any suitable manner such as being conveyed to
the interior of fan nose cone 12 via the hollow low pressure shaft
7 shown in FIG. 1. The shell of fan nose cone 12 may have opening
21 disposed in a forward end thereof through which the heated
pressurized air within the fan nose cone 12 may be discharged as
indicated by arrows in FIG. 4. Even though opening 21 is shown as a
single central opening in FIGS. 3 and 4, it is understood that,
alternatively, opening 21 could comprise a plurality of apertures.
For example, such apertures could be symmetrically distributed
about a central axis CL of fan 2, which may, in some embodiments,
correspond to a central axis of engine 10, to create the required
heated air curtain or boundary layer over the outer surface of fan
nose cone 12.
Deflector 22 may be configured as an axially symmetric plug that is
disposed forward (i.e., upstream) of the opening 21 and can be
supported in place by radial ribs, a central post or other
structures (not shown) of suitable external aerodynamic shape. The
deflector 22 may have a rearward surface 25 disposed forward and
spaced apart from the convex external surface of the shell to
thereby define an annular air flow channel 23 or slot. In various
embodiments, rearward surface 25 of deflector 22 may cooperate with
the convex external surface of fan nose cone 12 to form a
circumferentially-continuous annular flow channel 23.
Alternatively, such flow channel 23 may be non-continuous
(interrupted) due to radial ribs to form a plurality of apertures.
The axial position of the deflector 22 could also be adjustable
relative to fan nose cone 12 to change or regulate the flow of
heated air. The deflector 22 may have a central rearward convex
protrusion 24 that merges radially outwardly with a concave
rearward surface of revolution 25 that terminates at a
circumferential edge 26. In the illustrated example, the concave
rearward surface of revolution 25 has a partial toroid surface
although other shapes suitable to direct the air flow in a suitable
manner are possible.
The geometric relationship between the size of the opening 21,
curvature of the exterior surface of fan nose cone 12, curvature of
the concave rearward surface of revolution 25, and size of the
annular air flow channel 23 together with heated air pressure and
engine operating parameters may be selected to create a Coanda
effect and produce an attached air curtain or boundary layer of
heated air flowing downstream along the exterior surface of the fan
nose cone 12. For example, in some embodiments, the height H of the
annular gap provided by flow channel 23 and the radius of curvature
R of the exterior surface of fan nose cone 12 adjacent flow channel
23 may be related and selected to provide the Coanda effect. For
example, in some embodiments, a ratio between radius R and height H
(i.e., R/H) as illustrated in FIG. 4 may be selected based on a
Reynolds number of the flow through flow channel 23. For example,
in some embodiments, the ratio R/H may be inversely related (e.g.,
proportional) to the Reynolds number so that a higher ratio of R/H
may be suitable for a smaller Reynolds number. The ratio of R/H and
actual values of R and H may be selected based on testing (i.e.,
empirically) or based on numerical analysis and may depend on the
specific installation and operating conditions. In some
embodiments, the curvature of a portion of the external surface of
fan nose cone 12 adjacent flow channel 23 may have a generally
constant radius of curvature R so that the external surface may
have a generally arcuate cross-sectional profile in that particular
portion. In some embodiments, the portion of the external surface
may be outwardly curved according to a suitable polynomial
function. In some embodiments, different portions of the external
surface of fan nose cone 12 may have different curvatures.
The deflector 22 may have an exterior convex forward surface of
revolution 27 to direct incoming air flow. In some embodiments, the
exterior convex forward surface of revolution 27 may merge at the
circumferential edge 26 without a sharp edge. The concave rearward
surface of revolution 25 of deflector 22 together with incoming air
flowing over the exterior convex forward surface of revolution 27
may direct heated pressurized air being discharged from the opening
21 to flow in a manner forming a heated pressurized air curtain
over at least part of the exterior surface of the fan nose cone 12
in a radially outward and downstream direction as indicated by
arrows in FIG. 4.
In some embodiments, the substantial attachment of the heated air
boundary layer may extend an area of the fan nose cone 12 that can
be maintained above water freezing temperature at the lower
pressures encountered during flight and may improve anti-icing
capability. The longer that a heated air curtain is in contact or
maintained adjacent the outer surface of the fan nose cone 12, the
longer that area of the fan nose cone 12 may be protected from
icing.
In addition to anti-icing benefits, the above described deflector
22 and fan nose cone 12 arrangement may, in some embodiments,
improve engine efficiency due to axial momentum recovery. For
example, as opposed to discharging the flow of heated air from the
opening 21 directly in the upstream direction and in direct
opposition to the incoming air flow entering the fan 2 and engine
10, the use of deflector 22 causes the heated air to be directed in
a more favorable direction. The heated flow of air requires engine
power to create the necessary air pressure, upstream velocity and
temperature. The loss of axial momentum l.sub.0 for a situation
where the heated air would be discharged directly in the upstream
direction can be represented by the formula
l.sub.0=mass.times.axial velocity vector=-mv. Since the axial
velocity vector is negative (i.e., in an upstream direction), the
axial momentum l.sub.0 is also negative and would represent a loss
in momentum in such exemplary situation.
In contrast, referring to FIG. 4, the diverted or redirected flow
of heated pressurized air shown by arrows is in a radially outward
and downstream direction. The velocity vector has a radially
outward components which are equal, in opposite directions and are
balanced and therefore having no effect on axial momentum. On the
other hand, the axial component of the velocity vector is directed
downstream and hence is positive and is in a favorable direction.
Assuming that the mass of heated air flow (m) and velocity (v) when
discharged are identical to above example, the gain of axial
momentum l.sub.1 can be represented by the formula
l.sub.1=mass.times.axial velocity vector. Since the velocity vector
is positive (i.e., downstream direction), the axial momentum
l.sub.1 is also positive and represents a gain, which results in a
net gain in momentum .DELTA.l as follows: l.sub.1=mass.times.axial
velocity vector=m.times.air velocity(v).times.cosine .alpha..
where .alpha. is the angle of the velocity vector relative to the
engine axis which may correspond to the central axis of fan 2 in
some embodiments.
Accordingly the net gain in momentum .DELTA.l can be calculated as
follows: .DELTA.l=l.sub.0-l.sub.0=mvcos .alpha.-(-mv)=mv(cos
.alpha.+1).
Therefore in some embodiments, the penalty in a loss of momentum
l.sub.0 may be avoided and the axial downstream flow of air may add
to forward momentum l.sub.1 resulting in a net change or gain of
momentum .DELTA.l=m.about.v (cos .alpha.+1).
FIGS. 5 and 6 illustrate an optional circumferential vent 28
located at a downstream edge of the fan nose cone 12 that is
configured to discharge heated pressurized air to form a heated
pressurized air curtain or boundary layer flow from the vent 28 in
a radially outward and downstream direction to improve anti-icing
of the fan blade platforms 16 downstream from the circumferential
vent 28.
In some embodiments, the exterior of the fan nose cone 12 and fan
blade platforms 16 downstream from the tip of the fan nose cone 12
may, in some conditions, may be prone to icing formation if the air
temperature is sufficiently low. In order to provide an additional
heated air flow in these areas, the fan nose cone 12 may include an
optional rearward circumferential vent 28 in communication with the
source of heated pressurized air from the engine 10. FIG. 6 shows
with arrow 29 a flow of heated air moving from within the interior
of the fan nose cone 12 to an annular plenum 30. The presence of
heated air in contact with the interior surface of the fan nose
cone 12 may raise the temperature of the fan nose cone 12 and raise
the temperature of the exterior surface the fan nose cone 12 as
well. The pressurized heated air may then be discharged from the
vent 28 to form a heated pressurized air curtain in a radially
outward and downstream direction immediately upstream from the fan
blade platforms 16 (e.g., upstream from blades 1).
In the example shown in FIGS. 5 and 6, the circumferential vent 28
is disposed immediately upstream from fan blade platforms 16.
Additional intermediary vents (not shown) having a continuous
circumferential opening like vent 28 could also be disposed
anywhere along the length of the fan nose cone 12. Intermediary
vents would provide a heated curtain of air between the tip of the
fan nose cone 12 and the vent 28 to de-ice the intermediate areas
of the fan nose cone 12.
Although the above description relates to a specific preferred
embodiment as presently contemplated by the inventors, it will be
understood that the invention in its broad aspect includes
mechanical and functional equivalents of the elements described
herein.
* * * * *